Johannes Kepler (1571-1630) developed a quantitative description of the motions of the planets in the solar system. The description that he produced is expressed in three ``laws''.

*The orbit of a planet about the Sun is an ellipse with the Sun at
one focus.*

Figure 1 shows a picture of an ellipse. It is constructed by
specifying two *focus* points, F1 and F2, of the ellipse. All
points on the ellipse, such as P in Figure 1, have the property that
the sum of the distance between P and F1 and the distance between P and
F2 is a constant. The dimension of an ellipse is often described by
giving its *major axis* and *minor axis*. In
descriptions of orbits in the solar system, however, it is more common
to use the *semi-major axis* to describe the size of the orbit,
and the *eccentricity* of the ellipse to describe its shape.
The eccentricity is given by the ratio of the distance between the two
focus points to the length of the major axis of the ellipse. The periapsis,
or the shortest distance between the orbiting body and the central mass,
is determined by the product of the semi-major axis and the complement
of the eccentriciy (1 - e): if the body is orbiting the sun, this is the
peri*helion*, symbolized by q): q = a (1 - e). A circle
is a special case of an ellipse, with an eccentricity of 0, or so that q = a.

*A line joining a planet and the Sun sweeps out equal areas in equal
intervals of time.*

Figure 2 illustrates Kepler's Second Law. Consider the line between
the Sun and point A on the elliptical orbit. After a certain amount of
time, the planet will have moved along the orbit to point B, and the
line between the Sun and the planet will have swept over the cross
hatched area in the figure. Kepler's Second Law states that for
*any* two positions of the planet along the orbit that are
separated by the same amount of time, the area swept out in this manner
will be the same. Thus, suppose that it takes the planet the same
amount of time to go between positions C and D as it did for the planet
to go between positions A and B. Kepler's Second Law then tells us
that the second cross hatched area between C, D, and the Sun will be
the same as the cross hatched area between A, B, and the Sun.

Kepler's Second Law is valuable because it gives a quantitative
statement about how fast the object will be moving at any point in its
orbit. Note that when the planet is closest to the Sun, at
*perihelion*, Kepler's Second Law says that it will be moving
the fastest. When the planet is most distant from the Sun, at
*aphelion*, it will be moving the slowest.

*The squares of the sidereal periods of the planets are proportional to the cubes of their semimajor axes.*

We have defined the semimajor axis of the orbit above, in our
discussion of Kepler's First Law. The *sidereal period* of a
planet's orbit is the time that it takes a planet to complete one orbit
around the Sun. Kepler discovered a quantitative relationship between
these two properties of the orbit. If *P* is the period of the orbit,
measured in years, and *a* is the semimajor axis of the orbit, measured
in Astronomical Units, then

*P ^{2} = a^{3}*

Kepler's Laws are wonderful as a *description* of the motions of
the planets. However, they provide no explanation of *why* the
planets move in this way. Moreover, Kepler's Third Law only works for
planets around the Sun and does not apply to the Moon's orbit around
the Earth or the moons of Jupiter. Isaac Newton (1642-1727) provided a
more general explanation of the motions of the planets through the
development of *Newton's Laws of Motion* and *Newton's
Universal Law of Gravitation.*

One way to describe the motion of an object it to specify its
*position* at different times. Consider the car in Figure 3.
We can tell where it is at different times as it travels down a road.
It starts at milepost 0. One minute later it is between mileposts 1
and 2 at a distance of about 1.3 miles from the start. Two minutes
later, the car has gotten to a distance of about 3.3 miles from the
start. In general, we could specify a unique position for the car at
any time. For example, we might have written down where the car was at
a time 1.5 minutes after the start, and even if we hadn't, we're pretty
sure that the car was, in fact, *somewhere*. Mathematicians call
this kind of a relationship a *function*. When we say that the
position of the car is a function of time, it just means that there is
a unique location for the car for any time. For a planetary orbit, we
can describe the orbit in the same way, by providing the position of
the planet along the orbit for all times.

Another useful property for describing motion is the *velocity* of
the object. Velocity is defined to be the change of position with
change in time. Thus, for our car moving along the road, we can find
the velocity by dividing the distance travelled by the time it takes to
travel that distance. In our example, during the first minute, the car
travels 1.3 miles along the road. Thus, the car's velocity would be
1.3 miles per minute (or about 78 miles per hour!) on the average
during that first minute. It is important to note that physicists are
very particular about the definition of velocity, and when we state a
velocity we always make a statement about the *direction* of the
motion. In our one dimensional case, this corresponds to my statement
that the the car moved along the road. In general, if we were looking
at a road map, we might say that the velocity was 1.3 miles per minute
towards the East -- if the street ran towards the East. Velocity
always is specified by both a value and a direction.

A final useful property for describing motion is the *acceleration*
of the object. Just as the velocity describes the rate of change in
the position of the object, the acceleration describes the rate of
change of the velocity. In our example, the car moved farther during
its second minute of travel than it did during its first minute. The
average velocity during the second minute would be 2 miles per minute
(120 miles per hour), since the car covered two miles from 1.3 to 3.3
during the one-minute time interval from 1 minute after the start to 2
minutes after the start. The velocity increased a lot (0.7 miles per
minute) between the first minute of travel and the second minute of
travel, and we describe this change by the acceleration. In this case,
the car's velocity increased by 0.7 miles per minute in a time interval
of one minute. Thus, we'd say that the average acceleration of the car
during this time was 0.7 miles per minute PER MINUTE --- acceleration is
the rate of change of the velocity.

Like velocity, acceleration has both a value and a direction implied.
In our example, the direction was ``along the road'', but in a more
general case, the acceleration is not necessarily in the same direction
as the velocity. An especially good example for understanding the
solar system is the case of uniform circular motion. Lets consider the
case below of a car moving around a circle. The *speed* is
constant in this motion, but the *direction* is changing
continuously -- note the arrows showing the direction of motion in the
figure -- so there must be an acceleration here. The acceleration in
this special case of circular motion is called the *centripetal
acceleration*. It is always in the direction of the center of the
circle, as indicated in the figure, and it has a value, *A*, of

*A = v ^{2} / R*

where *v* is the speed of the object along its circular path,
and *R* is the radius of the circle.

*A body remains at rest or moves in a straight line at a constant speed unless it is acted upon by an outside force.*

If you look back at the definition of acceleration, you will see that:
(1) a body at rest is not accelerating; and (2) a body moving in a
straight line at a constant speed is not accelerating either. Thus,
the first law of Newton says that objects do not accelerate
*unless* they are acted upon by an outside force.

*If a force, F, works on a body of mass M, then the acceleration, A, is given by*

*F = M A*

The first law said that if there is acceleration, then there is a
force. Newton's second law gives a quantitative relationship between
the force and the acceleration that is observed. The relationship
depends on a new property of the object, its *mass*. The mass is
simply a measure of the amount of material in the object; mass is
conventionally measured in grams or kilograms. Note that the second
law implies that, for a given force, a less massive body will
accelerate more than a more massive body. This is consistent with the
world you are familiar with. Shove your kid brother, he might move a
long way; shove Shaquille O'Neal with the same force
and he won't move that far...

*If one body exerts a force on a second body, the second body
exerts an equal and opposite force on the first.*

This law is sometimes called the ``Action-Reaction'' law. Consider what happens if you are in one row boat and you pull on a line attached to a second row boat. When you pull the line, you exert a force on the second boat. But, by the third law, the other boat exerts an equal and opposite force back on you. Thus, if the second row boat has a large shipment of bricks in it so it is very heavy, your lighter boat may do all the moving even though you are doing all the pulling.

The elliptical orbits of the planets have such small eccentricities that, to a very good approximation, we can think of them as circles. (Only very precise measurements, like those available to Kepler, are able to detect the difference.) This means that we can use the idea of uniform circular motion to analyze planetary motion. In that section, we revealed that a body in uniform circular motion was constantly accelerating towards the center of its circular track. Thus, according to Newton's first law of motion, there must be a force acting on the planet that is always directed toward the center of the orbit -- that is toward the Sun!

Newton's second law of motion allows us to state what the magnitude of
that force must be. The required force is just the mass of the Earth
times its acceleration. We know that the acceleration of an object
moving in uniform circular motion is A = V^{2}/R. Thus, we can
calculate the force that is *required* to keep the Earth on its
circular path and compare it to physical theories about what that force
might be. This is what Newton later did, although he did it first for
the Moon rather than the Earth, to learn about the force of Gravity.

Finally, let us consider an implication of the ``action-reaction'' law. If there is a force that attracts the Earth toward the Sun, then there must be an equal and opposite force attracting the Sun towards the Earth. Why, then, doesn't the Sun move? The answer is that it does move, but by a very small amount since the mass of the Sun is about half a million times that of the Earth. Thus, when subjected to the equal and opposite force required by the third law, it accelerates about half a million times less than the Earth as well. For this reason, to a very good approximation, we can treat the Sun as stationary in our studies of planetary motion.

By now you must be wondering: ``What is the Force that keeps the Earth
going around the Sun?'' Newton's great discovery was the force of {\sl
gravity}, which is an attractive force that occurs between two masses.
The *Universal Law of Gravitation* is usually stated as an
equation:

*F _{gravity} = G M_{1} M_{2} / r^{2}*

where *F _{gravity}* is the attractive gravitational force between two
objects of mass

*G = 6.67 X 10 ^{-11} meters^{3} kilograms^{-1} seconds^{-2}*

Newton's great step was developing this law and using it, with his laws
of motion, to explain the motion of *lots* of different things ---
from falling objects to planets. Amazingly, out of these simple and
general rules, Newton was able to show that all of Kepler's
descriptive laws for orbits followed as a direct consequence.

When you combine Newton's gravitation and circular acceleration, which
must balance in order for the object to remain in orbit, you get a nice
relation between the period, distance, and mass of the central body.
It beings by equating the centripetal force (F_{cent})
due to the circular motion to the gravitational force (F_{grav}):
F_{grav} = F_{cent}

F_{grav} = G m_{1} m_{2} / r^{2}

F_{cent} = m_{2} V^{2} /r

Let the Earth be m_{1} and the Moon be m_{2}. For
circular motion the distance r is the semi-major axis a. The orbital
velocity of the Moon can be described as distance/time, or circumference
of the circular orbit divided by the orbital period:

V = 2 pi r /P

so setting the forces equal yields

G m_{1} m_{2} / a^{2} = m_{2} V^{2} /a

note that the m_{2} will cancel, so that circular orbital motion is independent of the mass of the orbiting body!

G m_{1} / a^{2} = ((2 pi a)^{2}/P^{2})/a

which we rearrange to place all the a-terms on the right and all the P-terms on the left:

G m_{1}/(4 pi^{2}) P^{2} = a^{3}

which should look startlingly like Kepler's third law, but this time for the Earth's mass (or any other) instead of the sun's mass. To use a and P to solve for mass, manipulate once more so that

m_{1} = a^{3} (4 pi^{2}/G) / P^{2}

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